All attempts failed so far to represent the Two proofs of the RH (P1, P2) are given based on spectral theory in the framework of distributional Hilbert spaces and two appropriately built self adjoint integral operators with related domains. The kernel functions of two convolution integral operators are built as
The Hilbert transform of the even Gaussian function f(x) is given by the odd Dawson function F(x) (which is the derivative of the square of the erf(x)-function) with its appreciated properties (see also "overview"). The Mellin transform M of the Gaussian function f(x) in combination with the singular "Zeta(s)" function (at s=1) defines Riemann's entire Zeta function "Z(s)" by Z(s) := s*(1-s)*Zeta(s)*M(f)(s) = Z(1-s). The key idea of
Z(s)=Z(1-s) based on an underlying self-adjoint integral operator.
P1 Hilbert space: L(2)-functions with real-axis domain P2 Hilbert space: periodic L(2)-functions with (0,1)-domain. Following this approach the correspondingly built distribution valued holomorphic function ((PeB) p. 38) does have all its zeros on the critical line. In combination with Hardy's theorem ((EdH) 11.1.) that there are infinite zeros on the critical line this proves the RH also in a strong sense.
last update (Note 34): April 1, 2015
last update: Jan 8, 2015
A distributional framework is proposed to leverage current Hardy-Littlewood circle method enabling a proof of both, the binary and ternary Goldbach conjectures. The latter one is proven for odd numbers N greater than c*exp(43000), only. The today´s Fourier analysis of Weyl (periodical, trigonometric) sums in a Banach space framework is proposed to be replaced by a discrete wavelet analysis on the circle in a distributional Hilbert space framework based on the (hypergeometric, non-periodical) Kummer function (WIP). last update: Jan 8, 2015
The spectral theory based argument can be applied to both Theta functions, based on the Gauss-Weierstrass function, as well as to the fractional part function. The Hilbert transforms of the Gauss-Weierstrass function and the fractional part function define appropriate kernels of integral operators with corresponding domains and ranges, embedded in appropriate Hilbert spaces:
The corresponding orthogonal systems of the related Hilbert spaces enable the definition of alternative Fourier integral representations of the Zeta function itself. For the integral operators of P1 and P2 the following polynomial systems are proposed: P1: The Hilbert transformed Hermite polynomials P2: The Lommel and/or Bessel polynomials.
The constant not vanishing terms of the Theta functions are the root cause of only formally self-adjoint invariant operator definitions with corresponding transform of the Riemann duality equation ((EdH), 10.3). This fact is also reflected in the assumptions of the Müntz formula ((TiE) 2.11). The formula requires additional convergence conditions of the "integral density" function in order to guarantee, that the "inversion ("dual"-1/x variable substitution) operation" is justified. The Poisson summation of the even Gauss-Weierstrass function does not fulfill those conditions. Therefore the Müntz formula cannot be applied to build the Riemann duality equation as transform of an appropriately defined integral operator. The root cause for this is that the Müntz formula is valid in the Banach space framework of the continuous functions equipped with the max-norm.
Distributional Hilbert spaces are "elements" of the Hilbert scale H(a) with a<.0. In case of elements u and v of H(0)=L(2) (a=0) with same norm, u and v are identical in a weak H(0) sense, as the L(2) Hilbert space is isomorph to its dual (distribution) space. The Hilbert transform has the following properties: - the Hilbert transform of a L(2)-function is again a L(2)-function - H(b) is densely embedded into H(a) (a<b) with respect to the norm of H(a).
For the relationship between the Theta function property and the Riemann duality equation we refer to:
the function f(x) fulfills the Theta property iff M(f)(s)=M(f)(1-s).
if f fulfills the Theta property, then g fulfills the Theta property in a weak L(2) sense and the constant Fourier term of g is absent.
As a consequence it holds:
Then, by density arguments in combination with Hardy's theorem the RH is also true in a strong sense.
and the references cited there.
The above solution concept puts the degenerated hypergeometric (Kummer) functions on the stage. The asymptotics of its zeros can be found in
The positive zeros of the Bessel functions (Polya G., "Ueber einen Satz von Laguerre") are related to a positive representation of polynomials (Polya G., "Ueber positive Darstellung von Polynomen") by the Jensen criterion:
G. Polya based his analysis of zeros of certain entire function on the Theorem of Kakeya, that each polynomial with positive and increasing coefficients have only zeros with absolute value less than 1:
With respect to the Plancherel-Polya theorem we refer to
There is a relationship between a real valued periodic function f(x), its number N(k) of changes of sign of its derivatives in a period and the Hilbert space H(-a), a>0, that f(x) is an entire function:
Let INTEGRAL(f) denote the integral of a real valued function f(x) from x=0 to x=infinite, g(x) be the conjugate of f(x) and x*F(x):=g(1/x). Let further denote M the Mellin transform operator. The multiplicative convolution of f(x) and h(x) is defined by (f * h)(x):=INTEGRAL(f(x/y)h(y)dy/y. Then for the Mellin transform of f * F it holds:
The criterion for an appropriate function f(x) given in the paper from E. Bombieri and C. Lagardias ensures those convergent integrals in a strong (continuous function) framework. The concept and theorems of analytical representations of
This also addresses the conceptual issue, when trying to build the continuous analog (in a strong sense) of
We note, that the representation of this analytical function is in line with the pre-requisites for the "concluding remarks" of the paper of E. Bombieri, J. Lagardias.
Duffin R. J., Hilbert transforms in Yukawan Potential Theory, Proc. Nat. Acad. Sci. Vol. 69, No. 12, pp. 3677-3679
This relates to the sections "2nd proof, Jan 2011" and "Quantum Gravity", especially with respect to an alternative modeling of the ground state energy and its related opportunities to a modified quantum mechanics model.
As an
and just for fun:
Fourier analysis/inversion, invariant operators and their transforms, self-adjoint operators with transform Zeta function and the functional equation is given in Edwards H. M.:
zeros on an integral function represented by Fourier´s integral" is given inTitchmarsh E. C., "The Theory of the Riemann Zeta-function", Oxford University Press Inc., New York, 1951 (2.1-2.16, 10.23-10.24):
Titchmarsh E. C., The Theory of the Riemann Zeta-function, 2.11, A general formula involving Zeta(s).pdf [ 381,4 KB ]
Some related formulas concerning Bessel, Gamma and Zeta functions are summarized in:
The
The " A modified Voronoi summation formula with quadratic summands would fit to the corresponding representation of the quadrat of the Zeta function Z(s) (i.e. Z(s)*Z(s)) and its related infinite series representation with summands defined by the product of the classical divisor function d(n) and the "Zeta infinite series summands" n*exp(-s). | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||